<!DOCTYPE html>
<html lang="zh-CN">
<head>
  <meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=2">
<meta name="theme-color" content="#222">
<meta name="generator" content="Hexo 6.0.0">
  <link rel="apple-touch-icon" sizes="180x180" href="/images/apple-touch-icon-next.png">
  <link rel="icon" type="image/png" sizes="32x32" href="/images/favicon-32x32-next.png">
  <link rel="icon" type="image/png" sizes="16x16" href="/images/favicon-16x16-next.png">
  <link rel="mask-icon" href="/images/logo.svg" color="#222">

<link rel="stylesheet" href="/css/main.css">

<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Noto Serif SC:300,300italic,400,400italic,700,700italic|Roboto Mono:300,300italic,400,400italic,700,700italic&display=swap&subset=latin,latin-ext">
<link rel="stylesheet" href="/lib/font-awesome/css/all.min.css">
  <link rel="stylesheet" href="/lib/pace/pace-theme-bounce.min.css">
  <script src="/lib/pace/pace.min.js"></script>

<script id="hexo-configurations">
    var NexT = window.NexT || {};
    var CONFIG = {"hostname":"zhiruozzy.cn","root":"/","scheme":"Gemini","version":"7.8.0","exturl":false,"sidebar":{"position":"left","display":"post","padding":18,"offset":12,"onmobile":false},"copycode":{"enable":true,"show_result":true,"style":"default"},"back2top":{"enable":"truw","sidebar":true,"scrollpercent":true},"bookmark":{"enable":false,"color":"#222","save":"auto"},"fancybox":false,"mediumzoom":false,"lazyload":false,"pangu":false,"comments":{"style":"tabs","active":null,"storage":true,"lazyload":false,"nav":null},"algolia":{"hits":{"per_page":10},"labels":{"input_placeholder":"Search for Posts","hits_empty":"We didn't find any results for the search: ${query}","hits_stats":"${hits} results found in ${time} ms"}},"localsearch":{"enable":true,"trigger":"auto","top_n_per_article":1,"unescape":false,"preload":false},"motion":{"enable":true,"async":false,"transition":{"post_block":"fadeIn","post_header":"slideDownIn","post_body":"slideDownIn","coll_header":"slideLeftIn","sidebar":"slideUpIn"}},"path":"search.xml"};
  </script>

  <meta name="description" content="离散的一些习题和我的笔记（可恶的逻辑学，咬牙切齿">
<meta property="og:type" content="article">
<meta property="og:title" content="离散数学の笔记">
<meta property="og:url" content="http://zhiruozzy.cn/2022/03/30/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/index.html">
<meta property="og:site_name" content="芷若">
<meta property="og:description" content="离散的一些习题和我的笔记（可恶的逻辑学，咬牙切齿">
<meta property="og:locale" content="zh_CN">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323084335490.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323171729779.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323171721234.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220302221948240.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220302222040684.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323164327707.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323164434009.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323164512564.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326172328979.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326173308683.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326173448446.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326173842476.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326174225108.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326180354192.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326180948677.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326181140453.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326181608529.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326182456376.png">
<meta property="og:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326183553084.png">
<meta property="article:published_time" content="2022-03-30T05:15:52.000Z">
<meta property="article:modified_time" content="2022-04-07T03:32:53.867Z">
<meta property="article:author" content="芷若">
<meta property="article:tag" content="笔记">
<meta name="twitter:card" content="summary">
<meta name="twitter:image" content="http://zhiruozzy.cn/img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323084335490.png">

<link rel="canonical" href="http://zhiruozzy.cn/2022/03/30/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/">


<script id="page-configurations">
  // https://hexo.io/docs/variables.html
  CONFIG.page = {
    sidebar: "",
    isHome : false,
    isPost : true,
    lang   : 'zh-CN'
  };
</script>

  <title>离散数学の笔记 | 芷若</title>
  






  <noscript>
  <style>
  .use-motion .brand,
  .use-motion .menu-item,
  .sidebar-inner,
  .use-motion .post-block,
  .use-motion .pagination,
  .use-motion .comments,
  .use-motion .post-header,
  .use-motion .post-body,
  .use-motion .collection-header { opacity: initial; }

  .use-motion .site-title,
  .use-motion .site-subtitle {
    opacity: initial;
    top: initial;
  }

  .use-motion .logo-line-before i { left: initial; }
  .use-motion .logo-line-after i { right: initial; }
  </style>
</noscript>

<!-- hexo injector head_end start -->
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css">

<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/hexo-math@4.0.0/dist/style.css">
<!-- hexo injector head_end end --><link rel="stylesheet" href="/css/prism-tomorrow.css" type="text/css">
<link rel="stylesheet" href="/css/prism-line-numbers.css" type="text/css"></head>

<body itemscope itemtype="http://schema.org/WebPage">
  <div class="container use-motion">
    <div class="headband"></div>

    <header class="header" itemscope itemtype="http://schema.org/WPHeader">
      <div class="header-inner"><div class="site-brand-container">
  <div class="site-nav-toggle">
    <div class="toggle" aria-label="切换导航栏">
      <span class="toggle-line toggle-line-first"></span>
      <span class="toggle-line toggle-line-middle"></span>
      <span class="toggle-line toggle-line-last"></span>
    </div>
  </div>

  <div class="site-meta">

    <a href="/" class="brand" rel="start">
      <span class="logo-line-before"><i></i></span>
      <h1 class="site-title">芷若</h1>
      <span class="logo-line-after"><i></i></span>
    </a>
  </div>

  <div class="site-nav-right">
    <div class="toggle popup-trigger">
        <i class="fa fa-search fa-fw fa-lg"></i>
    </div>
  </div>
</div>




<nav class="site-nav">
  <ul id="menu" class="main-menu menu">
        <li class="menu-item menu-item-主页">

    <a href="/" rel="section"><i class="fa fa-home fa-fw"></i>主页</a>

  </li>
        <li class="menu-item menu-item-关于">

    <a href="/about/" rel="section"><i class="fa fa-user fa-fw"></i>关于</a>

  </li>
        <li class="menu-item menu-item-标签">

    <a href="/tags/" rel="section"><i class="fa fa-tags fa-fw"></i>标签</a>

  </li>
        <li class="menu-item menu-item-目录">

    <a href="/categories/" rel="section"><i class="fa fa-th fa-fw"></i>目录</a>

  </li>
        <li class="menu-item menu-item-归档">

    <a href="/archives/" rel="section"><i class="fa fa-archive fa-fw"></i>归档</a>

  </li>
      <li class="menu-item menu-item-search">
        <a role="button" class="popup-trigger"><i class="fa fa-search fa-fw"></i>搜索
        </a>
      </li>
  </ul>
</nav>



  <div class="search-pop-overlay">
    <div class="popup search-popup">
        <div class="search-header">
  <span class="search-icon">
    <i class="fa fa-search"></i>
  </span>
  <div class="search-input-container"></div>
  <span class="popup-btn-close">
    <i class="fa fa-times-circle"></i>
  </span>
</div>
<div class="algolia-results">
  <div id="algolia-stats"></div>
  <div id="algolia-hits"></div>
  <div id="algolia-pagination" class="algolia-pagination"></div>
</div>

      
    </div>
  </div>

</div>
    </header>

    
  <div class="reading-progress-bar"></div>

  <a href="https://github.com/zhiruozzy" class="github-corner" title="Follow me on GitHub" aria-label="Follow me on GitHub" rel="noopener" target="_blank"><svg width="80" height="80" viewBox="0 0 250 250" aria-hidden="true"><path d="M0,0 L115,115 L130,115 L142,142 L250,250 L250,0 Z"></path><path d="M128.3,109.0 C113.8,99.7 119.0,89.6 119.0,89.6 C122.0,82.7 120.5,78.6 120.5,78.6 C119.2,72.0 123.4,76.3 123.4,76.3 C127.3,80.9 125.5,87.3 125.5,87.3 C122.9,97.6 130.6,101.9 134.4,103.2" fill="currentColor" style="transform-origin: 130px 106px;" class="octo-arm"></path><path d="M115.0,115.0 C114.9,115.1 118.7,116.5 119.8,115.4 L133.7,101.6 C136.9,99.2 139.9,98.4 142.2,98.6 C133.8,88.0 127.5,74.4 143.8,58.0 C148.5,53.4 154.0,51.2 159.7,51.0 C160.3,49.4 163.2,43.6 171.4,40.1 C171.4,40.1 176.1,42.5 178.8,56.2 C183.1,58.6 187.2,61.8 190.9,65.4 C194.5,69.0 197.7,73.2 200.1,77.6 C213.8,80.2 216.3,84.9 216.3,84.9 C212.7,93.1 206.9,96.0 205.4,96.6 C205.1,102.4 203.0,107.8 198.3,112.5 C181.9,128.9 168.3,122.5 157.7,114.1 C157.9,116.9 156.7,120.9 152.7,124.9 L141.0,136.5 C139.8,137.7 141.6,141.9 141.8,141.8 Z" fill="currentColor" class="octo-body"></path></svg></a>


    <main class="main">
      <div class="main-inner">
        <div class="content-wrap">
          

          <div class="content post posts-expand">
            

    
  
  
  <article itemscope itemtype="http://schema.org/Article" class="post-block" lang="zh-CN">
    <link itemprop="mainEntityOfPage" href="http://zhiruozzy.cn/2022/03/30/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/">

    <span hidden itemprop="author" itemscope itemtype="http://schema.org/Person">
      <meta itemprop="image" content="/images/touxiang.jpg">
      <meta itemprop="name" content="芷若">
      <meta itemprop="description" content="">
    </span>

    <span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization">
      <meta itemprop="name" content="芷若">
    </span>
      <header class="post-header">
        <h1 class="post-title" itemprop="name headline">
          离散数学の笔记
        </h1>

        <div class="post-meta">
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="far fa-calendar"></i>
              </span>
              <span class="post-meta-item-text">发表于</span>

              <time title="创建时间：2022-03-30 13:15:52" itemprop="dateCreated datePublished" datetime="2022-03-30T13:15:52+08:00">2022-03-30</time>
            </span>
              <span class="post-meta-item">
                <span class="post-meta-item-icon">
                  <i class="far fa-calendar-check"></i>
                </span>
                <span class="post-meta-item-text">更新于</span>
                <time title="修改时间：2022-04-07 11:32:53" itemprop="dateModified" datetime="2022-04-07T11:32:53+08:00">2022-04-07</time>
              </span>
            <span class="post-meta-item">
              <span class="post-meta-item-icon">
                <i class="far fa-folder"></i>
              </span>
              <span class="post-meta-item-text">分类于</span>
                <span itemprop="about" itemscope itemtype="http://schema.org/Thing">
                  <a href="/categories/%E6%95%B0%E5%AD%A6%E7%AC%94%E8%AE%B0/" itemprop="url" rel="index"><span itemprop="name">数学笔记</span></a>
                </span>
            </span>

          <br>
            <span class="post-meta-item" title="本文字数">
              <span class="post-meta-item-icon">
                <i class="far fa-file-word"></i>
              </span>
                <span class="post-meta-item-text">本文字数：</span>
              <span>15k</span>
            </span>
            <span class="post-meta-item" title="阅读时长">
              <span class="post-meta-item-icon">
                <i class="far fa-clock"></i>
              </span>
                <span class="post-meta-item-text">阅读时长 &asymp;</span>
              <span>13 分钟</span>
            </span>

        </div>
      </header>

    
    
    
    <div class="post-body" itemprop="articleBody">

      
        <p>离散的一些习题和我的笔记（可恶的逻辑学，咬牙切齿</p>
<span id="more"></span>

<h3 id="习题"><a href="#习题" class="headerlink" title="习题"></a>习题</h3><h3 id="命题逻辑"><a href="#命题逻辑" class="headerlink" title="命题逻辑"></a>命题逻辑</h3><p>¬ ⟷ →Ⅴ ⋀⟺ ∀ ∃</p>
<h4 id="命题与联结词"><a href="#命题与联结词" class="headerlink" title="命题与联结词"></a>命题与联结词</h4><ol>
<li><p>x+y&gt;6；不是命题，因为其无法判断真假</p>
</li>
<li><p>“3大于或等于3”是复合命题，由简单命题“3大于3”和“3等于3”组成，联结词是：或</p>
</li>
<li><p>“张三和李四是同学”不是复合命题，因为“和”只是张三与李四的联结，不是两个命题的联结</p>
</li>
<li><p>a:这个课很有趣，b:这个习题很难 请表示：</p>
<p>这个内容有趣意味着这个习题很难，而且两者反之亦然：a⟷b</p>
<p>或者这个内容有趣，或者这个习题很难，并且两者恰具其一：(¬a ⋀b)Ⅴ(a ⋀¬b)</p>
</li>
<li><p>当且仅当：⟷，当两者真值相同时才为真</p>
</li>
<li><p>如果下雨，他就开车上班</p>
<p>只有下雨，他才开车上班</p>
<p>除非下雨，否则他不开车上班</p>
<p>上面三个句子符号化后都为p→q</p>
</li>
<li><p>不经一事，不长一智：</p>
<p>p:经一事 q:长一智  ¬p→¬q</p>
</li>
</ol>
<h4 id="判断真值"><a href="#判断真值" class="headerlink" title="判断真值"></a>判断真值</h4><ol>
<li>画真值表来判断（万能），A为重言式当且仅当A的真值表的最后一列全为1，A为矛盾式</li>
<li></li>
</ol>
<ul>
<li><strong>Let N(x) be the statement that “x has visited North Dakota,” where the domain consists of the students in</strong></li>
</ul>
<p><strong>your school. Express each of these quantifications in English.</strong> </p>
<p>a) ∃ x N(x) </p>
<p>Some student in the school has visited North Dakota. (Alternatively, there exists a student in the school who has </p>
<p>visited North Dakota.) </p>
<p>b) ∀ x N(x) </p>
<p>Every student in the school has visited North Dakota. (Alternatively, all students in the school have visited </p>
<p>North Dakota.) </p>
<p>c) ¬∃ x N(x) </p>
<p>This is the negation of part (a): No student in the school has visited North Dakota. (Alternatively, there does not </p>
<p>exist a student in the school who has visited North Dakota.) </p>
<p>d) ∃ x ¬N(x) </p>
<p>Some student in the school has not visited North Dakota. (Alternatively, there exists a student in the school who </p>
<p>has not visited North Dakota.) </p>
<p>e) ¬∀ x N(x) </p>
<p>This is the negation of part (b): It is not true that every student in the school has visited North Dakota. </p>
<p>(Alternatively, not all students in the school have visited North Dakota.) </p>
<p>f) ∀ x ¬N(x) </p>
<p>All students in the school have not visited North Dakota. (Author: This is technically the correct answer, </p>
<p>although common English usage takes this sentence to mean—incorrectly—the answer to part </p>
<ul>
<li><strong>Translate these statements into English, where R(x) is “x is a rabbit,” and H(x) is “x hops” and the domain</strong></li>
</ul>
<p><strong>consists of all animals.</strong> </p>
<p>a) ∀ x (R(x) → H(x)) </p>
<p>If an animal is a rabbit, then that animal hops. (Alternatively, every rabbit hops.) </p>
<p>b) ∀ x (R(x) ^ H(x)) </p>
<p>Every animal is a rabbit and hops (obviously not true). </p>
<p>c) ∃ x (R(x) → H(x)) </p>
<p>There exists an animal such that if it is a rabbit, then it hops. (Author: Note that this is trivially true, satisfied, </p>
<p>for example, by lions, so it is not the sort of thing one would say.) </p>
<p>d) ∃ x (R(x) ^ H(x)) </p>
<p>There exists an animal that is a rabbit and hops</p>
<ul>
<li><strong>Suppose that the domain of the propositional function P(x) consists of the integers−2,−1, 0, 1, and 2. Write out each of these propositions using disjunctions, conjunctions, and negations. （析取，连词和否定）</strong></li>
</ul>
<p>a) ∃xP(x)         b)∀xP (x)           c) ∃x¬P(x) </p>
<p>​     d) ∀x¬P(x)       e) ¬∃xP (x)         f) ¬∀xP (x) </p>
<p>Answers: </p>
<p>Existential quantifiers are like disjunctions, and universal quantifiers are like </p>
<p>conjunctions. （存在量词类似于析取，普遍量词类似于合取）</p>
<p>a)P (−2) ∨ P (−1) ∨ P (0) ∨ P (1) ∨ P (2) </p>
<p>b) P (−2) ∧ P (−1) ∧ P (0) ∧ P (1) ∧ P (2) </p>
<p>c) ¬P (−2) ∨ ¬P (−1) ∨ ¬P (0) ∨ ¬P (1) ∨ ¬P (2) </p>
<p>d) ¬P (−2) ∧ ¬P (−1) ∧ ¬P (0) ∧ ¬P (1) ∧ ¬P (2) </p>
<p>e)This is just the negation of part (a): ¬ ( P (−2) ∨ P (−1) ∨ P (0) ∨ P (1) ∨ P (2)) </p>
<p>f)This is just the negation of part (b): ¬ ( P (−2) ∧ P (−1) ∧ P (0) ∧ P (1) ∧ P (2)) </p>
<ul>
<li><strong>Express each of these system speciﬁcations using predicates, quantiﬁers, and logical connectives</strong></li>
</ul>
<p>a) At least one mail message, among the nonempty set of messages, can be saved if there is a disk with more than 10 kilobytes of free space.</p>
<p>&#x3D;&gt; (∃x F(x, 10)) →∃x S(x), where F(x, y) is “Disk x has more than y kilobytes of free space,” and S(x) is “Mail message x can be saved” </p>
<p> b) Whenever there is an active alert, all queued messages are transmitted. </p>
<p>&#x3D;&gt; (∃x A(x)) →∀x(Q(x) → T (x)), where A(x) is “Alert x is active,” Q(x) is “Message x is queued,” and T (x)is “Message x is transmitted”</p>
<p>c) The diagnostic monitor tracks the status of all systems except the main console. </p>
<p>&#x3D;&gt;∀x((x &#x3D; main console) → T (x)) , where T (x)is “The diagnostic monitor tracks the status of system x” </p>
<p>d) Each participant on the conference call whom the host of the call did not put on a special list was billed.</p>
<p>&#x3D;&gt;∀x(¬L(x) → B(x)), where L(x) is “The host of the conference call put participant x on a special list” and B(x)is “Participant x was billed” </p>
<ul>
<li><strong>Translate these speciﬁcations into English where F(p)is “Printer p is out of service,” B(p) is “Printer p is busy,” L(j) is “Print job j is lost,” and Q(j) is “Print job j is queued.”</strong></li>
</ul>
<p>a) ∃p(F(p)∧B(p))→∃jL(j) </p>
<p>&#x3D;&gt;If there is a printer that is both out of service and busy, then some job has been lost. </p>
<p>b) ∀pB(p) →∃jQ(j)</p>
<p>&#x3D;&gt; If every printer is busy, then there is a job in the queue.</p>
<p> c) ∃j(Q(j)∧L(j)) →∃pF(p) </p>
<p>&#x3D;&gt;If there is a job that is both queued and lost, then some printer is out of service.</p>
<p>d) (∀pB(p)∧∀jQ(j)) →∃jL(j) </p>
<p>&#x3D;&gt;) If every printer is busy and every job is queued, then some job is lost. </p>
<ul>
<li><p><strong>Translate to Logic</strong></p>
<p>Express each of these system specififications using predicate, quantififiers, and logical connectives.</p>
<p>(a) Every user has access to an electronic mailbox.</p>
<p><strong>Solution:</strong></p>
<p>Let the domain be users and mailboxes. Let User(<em>x</em>) be “<em>x</em> is a user”, let Mailbox(<em>y</em>) be “<em>y</em> is a mailbox”,</p>
<p>and let Access(<em>x, y</em>) be “<em>x</em> has access to <em>y</em>”.</p>
<p><strong>∀  x (User(x) → ( ∃ y (Mailbox(<em>y</em>)  ∧ Access(x, y))))</strong></p>
<p>(b) The system mailbox can be accessed by everyone in the group if the fifile system is locked.</p>
<p><strong>Solution:</strong></p>
<p>Let the domain be users and mailboxes. Let Access(<em>x, y</em>) be “<em>x</em> has access to <em>y</em>”. Let GroupMember(<em>x</em>)</p>
<p>be “<em>x</em> is a member of the group.” Let FileSystemLocked be the proposition “the fifile system is locked.” Let</p>
<p>SystemMailbox be the constant that is the system mailbox.</p>
<p><strong>FileSystemLocked <em>→ ∀ x</em> (GroupMember(<em>x</em>) <em>→</em> Access(<em>x,</em> SystemMailbox))</strong></p>
<p>(c) The fifirewall is in a diagnostic state only if the proxy server is in a diagnostic state.</p>
<p><strong>Solution:</strong></p>
<p>Let the domain be all applications. Let Firewall(<em>x</em>) be “<em>x</em> is the fifirewall”, and let ProxyServer(<em>x</em>) be “<em>x</em> is</p>
<p>the proxy server.” Let Diagnostic(<em>x</em>) be “<em>x</em> is in a diagnostic state”.</p>
<p><em>∀</em> <em>x</em> <em>∀  y</em> ((Firewall(<em>x</em>) <em>∧</em> Diagnostic(<em>x</em>)) <em>→</em> (ProxyServer(<em>y</em>) <em>→</em> Diagnostic(<em>y</em>))</p>
<p>(d) At least one router is functioning normally if the throughput is between 100kbps and 500 kbps and the</p>
<p>proxy server is not in diagnostic mode.</p>
<p><strong>Solution:</strong></p>
<p>Let the domain be all applications and routers. Let Router(<em>x</em>) be “<em>x</em> is a router”, and let ProxyServer(<em>x</em>)</p>
<p>be “<em>x</em> is the proxy server.” Let Diagnostic(<em>x</em>) be “<em>x</em> is in a diagnostic state”. Let ThroughputNormal be</p>
<p>“the throughput is between 100kbps and 500 kbps”. Let Functioning(<em>y</em>) be “y is functioning normally”.</p>
<p><strong><em>∀</em> <em>x</em> (ThroughputNormal <em>∧</em> (ProxyServer(<em>x</em>) <em>∧ ¬</em>Diagnostic(<em>x</em>))) <em>→</em> (<em>∃</em> <em>y</em> Router(<em>y</em>) <em>∧</em> Functioning(<em>y</em>))</strong></p>
</li>
<li><p>Translate these system specififications into English where <em>F</em>(<em>p</em>) is “Printer <em>p</em> is out of service”, <em>B</em>(<em>p</em>) is “Printer <em>p</em></p>
<p>is busy”, <em>L</em>(<em>j</em>) is “Print job <em>j</em> is lost,” and <em>Q</em>(<em>j</em>) is “Print job <em>j</em> is queued”. Let the domain be all printers together</p>
<p>with all print jobs.</p>
<p>(a) <em>∃</em> <em>p</em> (<em>F</em>(<em>p</em>) <em>∧</em> <em>B</em>(<em>p</em>)) <em>→ ∃</em> <em>j L</em>(<em>j</em>) </p>
<p>1<strong>Solution:</strong></p>
<p>If at least one printer is busy and out of service, then at least one job is lost.</p>
<p>(b) (<em>∀</em> <em>p B</em>(<em>p</em>)) <em>→</em> (<em>∃</em> <em>j Q</em>(<em>j</em>))</p>
<p><strong>Solution:</strong></p>
<p>If all printers are busy, then there is a queued job.</p>
<p>(c) <em>∃</em> <em>j</em> (<em>Q</em>(<em>j</em>) <em>∧</em> <em>L</em>(<em>j</em>)) <em>→ ∃</em> <em>p F</em>(<em>p</em>)</p>
<p><strong>Solution:</strong></p>
<p>If there is a queued job that is lost, then a printer is out of service.</p>
<p>(d) (<em>∀</em> <em>p B</em>(<em>p</em>) <em>∧ ∀</em> <em>j Q</em>(<em>j</em>)) <em>→ ∃</em> <em>j L</em>(<em>j</em>)</p>
<p><strong>Solution:</strong></p>
<p>If all printers are busy and all jobs are queued, then there is some lost job.</p>
</li>
<li><p><strong>Express each of these system specififications using predicates, quantififiers, and logical connectives.</strong></p>
<p><strong>a) Every user has access to an electronic mailbox.</strong></p>
<p><strong>Solution</strong>: We can limit the domain to all users of the system, and introduce the <em>E</em> predicate:</p>
<p><em>E</em>(<em>x, e</em>) :&#x3D; user <em>x</em> has access to electronic mailbox <em>e</em></p>
<p>Therefore, we’re quantifying over two domains in the <em>E</em> predicate, the domain of system users and the domain of</p>
<p>electronic mailboxes. This requires that we quantify over both domains:</p>
<p>​                                                                       <em>∀</em> <em>x</em> <em>∃</em> <em>e M</em>(<em>x, e</em>)</p>
<p><strong>b) The system mailbox can be accessed by everyone in the group if the fifile system is locked.</strong></p>
<p><strong>Solution</strong>: You may be tempted to create a predicate quantifying over all fifile systems here, but we’re referring to</p>
<p>a single fifile system, in the defifinite sense. It’s the fifile system state that is uncertain, and which therefore needs to</p>
<p>be quantifified over. The states it can have are: locked, unlocked, . . . (possible further states of which we have no</p>
<p>knowledge). So, we introduce the predicate</p>
<blockquote>
<p><em>F</em>(<em>s</em>) :&#x3D; the fifile system is in state <em>s</em></p>
<p>For users in the group, we have the predicate</p>
<p><em>G</em>(<em>x</em>) :&#x3D; user <em>x</em> is in the group</p>
<p>and for system mailbox access, we have</p>
<p><em>SM</em>(<em>x</em>) :&#x3D; user <em>x</em> can access the system mailbox</p>
</blockquote>
<p>Now to write the actual statement. We want to ensure that the user <em>x</em> is in the group, and we want the condition</p>
<p>that the fifile system is locked to hold before we assert that users in the group can access the system mailbox. This</p>
<p>leads us to</p>
<p>​                                                           <em>∀**x G</em>(<em>x</em>) <em>∧</em> <em>F</em>(locked) <em>→</em> <em>SM</em>(<em>x</em>)</p>
<p>Notice that “locked” is a constant, representing the “locked” state of the fifilesystem (indeed, there is no propositional</p>
<p>variable named “locked” we’ve quantifified over! Any variable we introduce must be quantifified).</p>
<p><strong>c) The fifirewall is in a diagnostic state only if the proxy server is in a diagnostic state.</strong></p>
<p><strong>Solution</strong>: The tricky part here is that the fifirewall has any number of states, of which we know nothing, and several</p>
<p>of them could be considered “diagnostic” states. Likewise with the proxy server. Again, we’re referring to the</p>
<p>fifirewall and the proxy each in the defifinite sense, so they should not be quantifified over with variables. Instead, let’s  quantify over the domain of fifirewall and proxy states, and let’s introduce a predicate to identify diagnostic states:</p>
<blockquote>
<p><em>D</em>(<em>x</em>) :&#x3D; state <em>x</em> is a diagnostic state</p>
<p>and let’s introduce the predicates <em>P</em> and <em>F</em>: </p>
<p><em>F</em>(<em>x</em>) :&#x3D; the fifirewall is in state <em>x</em> </p>
<p><em>P</em>(<em>x</em>) :&#x3D; the proxy is in state <em>x</em></p>
</blockquote>
<p>From this, I was immediately led to write</p>
<p>​                                                                       <em>∀</em> <em>x D</em>(<em>x</em>) <em>∧</em> (<em>F</em>(<em>x</em>) <em>→</em> <em>P</em>(<em>x</em>))</p>
<p>but this is wrong. Why? Because the original statement allows the fifirewall and the proxy server to be in distinct</p>
<p>diagnostic states, and the implication should still hold. The use of the same variable means that the proxy server is</p>
<p>in a diagnostic state if fifirewall is in that exact same diagnostic state. In other words, I succumbed to the mistake</p>
<p>described in the fifinal part of question 2, above. We need distinct variables for these states, and they must both be</p>
<p>diagnostic states:</p>
<p>​                                                               <em>∀</em> <em>x</em> <em>∀</em> <em>y D</em>(<em>x</em>) <em>∧</em> <em>D</em>(<em>y</em>) <em>∧</em> (<em>F</em>(<em>x</em>) <em>→</em> <em>P</em>(<em>y</em>))</p>
<p><strong>d) At least one router is functioning normally if the throughput is between 100 kbps and 500 kbps and the proxy service</strong></p>
<p>is not in diagnostic mode.</p>
<p><strong>Solution</strong>: Again, the proxy service can have a number of modes, and we create a predicate to assert that the proxy</p>
<p>service is in mode <em>m</em>: </p>
<blockquote>
<p><em>P</em>(<em>m</em>) :&#x3D; the proxy server is in mode <em>m</em></p>
<p>The remaining predicates we need are straightforward:</p>
<p><em>R</em>(<em>r</em>) :&#x3D; router <em>r</em> is functioning normally</p>
<p><em>T</em>(<em>m, n</em>) :&#x3D; the throughput is between <em>m</em> kbps and <em>n</em> kbps</p>
<p><em>D</em>(<em>m</em>) :&#x3D; <em>m</em> is the diagnostic mode</p>
</blockquote>
<p>Since we are quantifying on at least one router that is functioning normally, the use of <em>∃</em> suffiffiffices over the domain of</p>
<p>routers. Similarly to the previous question, then, the fifinal proposition is</p>
<p>​                                                     <em>∀</em> <em>m</em> <em>∃</em> <em>r D</em>(<em>m</em>) <em>∧</em> [(<em>T</em>(100*,* 500) <em>∧ ¬</em> <em>P</em>(<em>m</em>)) <em>→</em> <em>R</em>(<em>r</em>)]</p>
</li>
</ul>
<h3 id="一些英译中"><a href="#一些英译中" class="headerlink" title="一些英译中"></a>一些英译中</h3><p>compound proposition :复合命题</p>
<p>negation of statements: 陈述句的否定</p>
<p>De Morgan’s laws :德摩根定律</p>
<p>truth tables：真值表</p>
<p> absorption laws：吸收律</p>
<p>tautology：重言式，永真式</p>
<p>logical operator：逻辑运算符</p>
<h3 id="命题"><a href="#命题" class="headerlink" title="命题"></a>命题</h3><h5 id="概念："><a href="#概念：" class="headerlink" title="概念："></a>概念：</h5><ul>
<li>命题，是指具有唯一真值的陈述句</li>
<li>疑问句、祈使句、感叹句，因为无法判断真假，所以都不是命题</li>
<li>真用1或T来表示，假用0或F来表示，因为命题只有这两种真值，所以这种逻辑成为二值逻辑</li>
</ul>
<h5 id="例题："><a href="#例题：" class="headerlink" title="例题："></a>例题：</h5><ul>
<li>1+101&#x3D;110  在二进制下为真，在十进制下为假，真值不唯一，所以不是命题</li>
<li>别的星球上有生物  虽然现在不确定，但是这个问题是有客观答案的 ，并不以你我的意志而转移，所以是命题</li>
<li>全体立正！ 祈使句不是命题</li>
<li>天气多好啊！ 感叹句不是命题</li>
</ul>
<p><strong>注意</strong>：一个陈述句暂时不能确定真值，但到了一定时间就可以确定，与一个陈述句的真值不能唯一确定是不一样的</p>
<ul>
<li>x&gt;3  　x的取值范围不同，本句话的真假是不一样的，所以不是命题</li>
<li>2190年人类将移居火星   虽然现在无法确定，但是到了2190年就可以唯一确定本句话的真假，所以是命题</li>
</ul>
<h5 id="分类："><a href="#分类：" class="headerlink" title="分类："></a>分类：</h5><ol>
<li>原子命题：一个陈述句再也不能分解成更为简单的语句，则由它构成的命题成为原子命题</li>
<li>复合命题：由原子命题，命题联结词和圆括号组成</li>
</ol>
<h3 id="命题联结词"><a href="#命题联结词" class="headerlink" title="命题联结词"></a>命题联结词</h3><ol>
<li><p>否定联结词： ┐</p>
<p>┐p和p的真假是相反的</p>
</li>
<li><p>合取联结词：∧</p>
<p>p∧Q读作“P与Q”或者“P且Q”  </p>
<p>当且仅当P和Q同为真，命题P∧Q的真值才为真</p>
</li>
<li><p>析取联结词：∨</p>
<p>P∨Q读作“P或Q”</p>
<p>只要P、Q中有一个为真，命题P∨Q就为真</p>
</li>
<li><p>条件(蕴含)联结词：→</p>
<p>P→Q读作P条件Q，或者“若P则Q” “P仅当Q” “P是Q的充分条件”</p>
<p>只有当P的真值为真而Q的真值为假时，命题P→Q的真值为假，其余都为真</p>
<p>例子：</p>
<p>①只要天下雨，我就回家</p>
<p>②只有天下雨，我才回家</p>
<p>③除非天下雨，否则我不回家</p>
<p>④仅当天下雨，我才回家</p>
<p>解析：①强调的是如果天下雨了，那我就回家，②③④强调的是如果我回家了，那一定是天下雨了。 所以①可符号化为P→Q ，②③④可符号化为Q→P</p>
</li>
<li><p>双条件联结词：↔</p>
<p>P↔Q读作“P当且仅当Q”，只有两者同时为真或同时为假时，P↔Q才为真</p>
</li>
<li><p>异或（双条件非）联结词：⊕</p>
<p>当且仅当P和Q的真值不相同时，P⊕Q为T</p>
</li>
<li><p>与非联结词：↑</p>
<p>A↑B &#x3D;  ┐(A∧B)</p>
</li>
<li><p>或非联结词：↓</p>
<p>A↓B &#x3D;  ┐(A∨B)</p>
</li>
</ol>
<h3 id="命题公式"><a href="#命题公式" class="headerlink" title="命题公式"></a>命题公式</h3><p>命题公式由原子命题，命题联结词，圆括号构成，但是并不是由这三类符号组成的的任何符号串都能成为命题公式 ，合理的命题公式叫做合式公式</p>
<h5 id="定义"><a href="#定义" class="headerlink" title="定义"></a>定义</h5><p>合式公式是由下列规则生成的公式：</p>
<ol>
<li>单个原子公式是合式公式</li>
<li>若A是一个合式公式，那么（ ┐A）也是一个合式公式</li>
<li>若A、B是合式公式，则（A^B）、（A∨B）、（A→B）和（A↔B）都是合式公式</li>
<li>只有有限次的使用1，2和3生成的公式才是合式公式</li>
</ol>
<h5 id="约定："><a href="#约定：" class="headerlink" title="约定："></a>约定：</h5><ol>
<li><p>联结词的优先级从高到低是： ┐、∧、∨、→、↔</p>
</li>
<li><p>相同的联结词按从左到右的次序计算时，圆括号可以省略</p>
</li>
<li><p>最外层的圆括号也可以省略</p>
</li>
</ol>
<h3 id="命题的翻译和符号化"><a href="#命题的翻译和符号化" class="headerlink" title="命题的翻译和符号化"></a>命题的翻译和符号化</h3><h5 id="将自然语言符号化"><a href="#将自然语言符号化" class="headerlink" title="将自然语言符号化"></a>将自然语言符号化</h5><ol>
<li><p>小王边走边唱</p>
<p>P：小王走路  Q：小王唱歌  </p>
<p>&#x3D;&gt; P^Q </p>
</li>
<li><p>如果今天不下雨并且不刮风，我就去书店</p>
<p>P：今天下雨（不能设位今天不下雨，因为“今天不下雨”不是原子命题）</p>
<p>Q：今天刮风</p>
<p>R：我去书店 </p>
<p>&#x3D;&gt;（┐P∧ ┐Q）→R</p>
</li>
<li><p>小刚要么在学习，要么在玩游戏</p>
<p>P：小刚在学习</p>
<p>Q：小刚在玩游戏</p>
<p>&#x3D;&gt; (P∧ ┐Q)∨(┐P∧ Q)  </p>
<p>即：小刚在学习且没有玩游戏，或者小刚在玩游戏且没有学习</p>
</li>
<li><p>除非a能被2整除，否则a不能被4整除</p>
<p>P：a能被2整除</p>
<p>Q：a能被4整除</p>
<p>&#x3D;&gt;  ┐P →  ┐Q   即：a如果不能被2整除，则a不能被4整除</p>
<p>或者Q→P   即：如果a能被4整除，那么a一定可以被2整除</p>
</li>
<li><p>如果天不下雨，我们就去打篮球，除非班上有会</p>
<p>P：天下雨</p>
<p>Q：我们去打篮球</p>
<p>R：今天班上有会</p>
<p>&#x3D;&gt;（┐P∧ ┐R）→Q   即：我们打篮球的条件必须是天不下雨且班上没有会</p>
<p>或者：┐R→(┐P→Q）即：如果没有会，并且没有下雨，我们就去打篮球</p>
</li>
<li><p>离散数学无用且枯燥无味是不对的</p>
<p>P：离散数学是有用的<br>Q：离散数学是枯燥无味的</p>
<p>¬ ( ¬ P ∧ Q ) </p>
</li>
<li><p>如果校长和小王都不去，则小李去</p>
<p>P：小张去<br>Q：小王去<br>R：小李去</p>
<p>( ¬ P ∧ ¬ Q ) → R </p>
</li>
<li><p>p，q不能同时取,即只能取一个</p>
<p>(P∧ ┐Q)∨(┐P∧ Q) </p>
</li>
<li><p>若P去, 则Q不能去</p>
<p>P →  ┐Q</p>
</li>
<li><p>C和D要么都有，要么都没有</p>
</li>
</ol>
<p>   (C∧D)∨(┐C∧ ┐D)</p>
<ol start="11">
<li><p>股票P和Q中必然有一种或两种要抛出</p>
<p>P∨Q</p>
</li>
</ol>
<h3 id="真值表"><a href="#真值表" class="headerlink" title="真值表"></a>真值表</h3><p>含有n个原子命题的命题公式所对应的真值共有2^n中情况</p>
<h3 id="等价"><a href="#等价" class="headerlink" title="等价"></a>等价</h3><h5 id="定义-1"><a href="#定义-1" class="headerlink" title="定义"></a>定义</h5><p>给定两个命题公式，若对于其中任意一组指派而言，A和B的真值都相同，则称A和B是等价的</p>
<p>可以通过等值关系进行化简得到两个式子等值，也可以通过真值表来判断等值</p>
<h5 id="基本等值式"><a href="#基本等值式" class="headerlink" title="基本等值式"></a>基本等值式</h5><ol>
<li><p>﹁ ( ﹁ G ) &#x3D; G<br>（双重否定律）</p>
</li>
<li><p>G ∧ G &#x3D; G   G ∨ G &#x3D; G  <strong>(a*a&#x3D;a  a+a&#x3D;a)</strong><br>（幂等律）</p>
</li>
<li><p>G ∨ H &#x3D; H ∨ G   <strong>a+b&#x3D;b+a</strong><br>G ∧ H &#x3D; H ∧ G  ab&#x3D;ba<br>（交换律）</p>
</li>
<li><p>G ∨ ( H ∨ S ) &#x3D; ( G ∨ H ) ∨ S  <strong>a+(b+c)&#x3D;(a+b)+c</strong><br>G ∧ ( H ∧ S ) &#x3D; ( G ∧ H ) ∧ S  *<em>a</em>(b <em>c)&#x3D;(a * b)<em>c</em></em><br>（结合律）</p>
</li>
<li><p>G ∨ ( H ∧ S ) &#x3D; ( G ∨ H ) ∧ ( G ∨ S )   a+bc&#x3D;(a+b)(a+c)<br>G ∧ ( H ∨ S ) &#x3D; ( G ∧ H ) ∨ ( G ∧ S )  <strong>a(b+c)&#x3D;ab+ac</strong><br>（分配律）</p>
</li>
<li><p>﹁ ( G ∧ H ) &#x3D; ﹁ G ∨ ﹁ H </p>
<p>﹁ ( G ∨ H ) &#x3D; ﹁ G ∧ ﹁ H </p>
<p> （去掉括号后合取变析取，析取变合取）<br>（德摩根律）</p>
</li>
<li><p>G ∨ ( G ∧ H ) &#x3D; G  <strong>a+ab&#x3D;a</strong></p>
<p>G ∧ ( G ∨ H ) &#x3D; G  <em><em>a</em>(a+b)&#x3D;a</em>*<br>（吸收律）</p>
</li>
<li><p>G ∨ 1 &#x3D; 1<br>G ∧ 0 &#x3D; 0<br>（零律）</p>
</li>
<li><p>G ∧ 1 &#x3D; G</p>
<p>G ∨ 0 &#x3D; G</p>
<p>（同一律）</p>
</li>
<li><p>G ∧﹁ G &#x3D; 0<br>G V ﹁ G &#x3D; 1<br>（否定律）</p>
</li>
<li><p>G → H &#x3D; ﹁ G ∨ H<br>（条件转化律）</p>
</li>
<li><p>G ↔ H &#x3D; ( G → H ) ∧ ( H → G ) &#x3D; ( ﹁ G ∨ H ) ∧ ( ﹁ H ∨ G )<br>（双条件传化律）</p>
</li>
<li><p>G → H &#x3D; ﹁ H → ﹁ G<br>（假言易位）</p>
</li>
<li><p>G ↔ H &#x3D; ﹁ G ↔ ﹁ H<br>（等价否定等式）</p>
</li>
</ol>
<h5 id="等价置换定理"><a href="#等价置换定理" class="headerlink" title="等价置换定理"></a>等价置换定理</h5><p>如果X是合式公式A的一部分，且X本身也是一个合式公式，则称X为合式公式A的子公式</p>
<p>设X是合式公式A的子公式，若X&#x3D;Y。如果将A中用的X用Y来置换，所得到的公式B与公式A等价。即A&#x3D;B</p>
<h3 id="重言式-永真式-与蕴含式-永假式"><a href="#重言式-永真式-与蕴含式-永假式" class="headerlink" title="重言式(永真式)与蕴含式(永假式)"></a>重言式(永真式)与蕴含式(永假式)</h3><h5 id="定义-2"><a href="#定义-2" class="headerlink" title="定义"></a>定义</h5><p>对于某个命题公式，如果其在分量的任何指派的指派下的真值均为T，则称该命题公式为<strong>重言式</strong>或<strong>永真式</strong>。</p>
<p>对于某个命题公式，如果其在分量的任何指派的指派下的真值均为F，则称该命题公式为<strong>矛盾式</strong>或<strong>永假式</strong>。</p>
<p>如果某个命题不是矛盾式，则称该命题为<strong>可满足式</strong></p>
<p><strong>蕴含式</strong>：当且仅当P→Q是重言式时，我们称“P蕴含Q“，并记作P&#x3D;&gt;Q</p>
<p>要证明P&#x3D;&gt;Q,只需证明P→Q,或者证明﹁Q→﹁P</p>
<h3 id="范式"><a href="#范式" class="headerlink" title="范式"></a>范式</h3><p>范式是析取范式与合取范式的总称。</p>
<h5 id="定义："><a href="#定义：" class="headerlink" title="定义："></a>定义：</h5><ul>
<li>命题变元或命题变元的否定称为<strong>文字</strong>。 P，﹁P，Q，﹁Q</li>
<li><em>有限个</em>文字的析取称为<strong>简单析取式</strong>（或<strong>子句</strong>）。 P(退化的析取式，只有一个文字），﹁P（退化的析取式）， P ∨ ﹁Q（2个文字）</li>
<li><em>有限个</em>文字的合取称为<strong>简单合取式</strong>（或<strong>短语</strong>）。 P，﹁P，P ∧ ﹁Q </li>
<li>单个的文字既是简单析取式，也是简单合取式</li>
<li>P与﹁P称为<strong>互补对</strong></li>
<li><em>有限个</em>简单合取式（短语）的析取式称为<strong>析取范式</strong>  如  ( G ∧ H ) ∨ ( G ∧ S )，又如 P ∨ ﹁Q ,P ,﹁P。<strong>内部合取，外部析取</strong></li>
<li><em>有限个</em>简单析取式（短语）的合取式称为<strong>合取范式</strong>，如（P∨ H ) ∧ ( ﹁H ∨Q）,又如 P ∧ ﹁Q ,P ,﹁P 。<strong>内部析取，外部合取</strong></li>
</ul>
<p><strong>注意1：p∧q∧r既是析取范式，也是合取范式</strong>。</p>
<p><strong>原因</strong>：因为这个式子是有限个文字的合取，所以他是合取范式，也是简单合取式。而根据析取范式的定义是有限个简单合取式的析取式，这个有限个可以取1个，即一个简单合取式也是析取式</p>
<h5 id="注意2："><a href="#注意2：" class="headerlink" title="注意2："></a>注意2：</h5><p>析取范式、合取范式仅含联结词集{﹁，∧，∨}，且否定联结词仅出现在命题变元之前</p>
<h5 id="范式存在定理"><a href="#范式存在定理" class="headerlink" title="范式存在定理"></a>范式存在定理</h5><p>对于任意公式，都存在与其等价的析取范式和合取范式，且范式不唯一</p>
<h5 id="转化方法："><a href="#转化方法：" class="headerlink" title="转化方法："></a>转化方法：</h5><ol>
<li><p>将公式中的↔，→ 用联结词﹁，∧，∨来取代（<strong>否定也是联结词</strong>）</p>
<ul>
<li><p>蕴含式：G → H &#x3D; ﹁ G ∨ H</p>
</li>
<li><p>双条件传化律：G ↔ H &#x3D; ( G → H ) ∧ ( H → G )</p>
</li>
</ul>
<p>利用蕴含式再次转化： &#x3D; ( ﹁ G ∨ H ) ∧ ( ﹁ H ∨ G )</p>
</li>
<li><p>将否定联结词内移(德摩根律)到各个命题变元的前端，并消去否定号（双重否定律）</p>
<ul>
<li><p>双重否定律：﹁(﹁ G)&#x3D;G</p>
</li>
<li><p>德摩根律：﹁ ( G ∧ H ) &#x3D; ﹁ G ∨ ﹁ H </p>
<p>​                  ﹁ ( G ∨  H ) &#x3D; ﹁ G ∧﹁ H </p>
<p>（去掉括号后析取变合取，合取变析取）</p>
</li>
</ul>
</li>
<li><p>利用分配律，将公式化成一些合取式的析取，或化成一些析取式的合取：</p>
<ul>
<li>分配律：G ∨ ( H ∧ S ) &#x3D; ( G ∨ H ) ∧ ( G ∨ S ) （合取范式）<br>          G ∧ ( H ∨ S ) &#x3D; ( G ∧ H ) ∨ ( G ∧ S ) （析取范式）</li>
</ul>
</li>
</ol>
<h5 id="例题：-1"><a href="#例题：-1" class="headerlink" title="例题："></a>例题：</h5><p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323084335490.png" alt="image-20220323084335490"></p>
<h3 id="主范式"><a href="#主范式" class="headerlink" title="主范式"></a>主范式</h3><p>因为范式的不唯一，为了规范化，形成唯一的主析取范式和主合取范式</p>
<h5 id="化成主范式的步骤："><a href="#化成主范式的步骤：" class="headerlink" title="化成主范式的步骤："></a>化成主范式的步骤：</h5><ul>
<li><p>先求出析取范式（合取范式）</p>
</li>
<li><p>将不是极小项（极大项）的简单合取式进一步化</p>
</li>
<li><p>极大项（极小项）用名称mi（Mi)表示，并用角标从小到大排序</p>
</li>
</ul>
<h5 id="极小项和极大项"><a href="#极小项和极大项" class="headerlink" title="极小项和极大项"></a>极小项和极大项</h5><p><strong>定义：</strong></p>
<p>在含有n个命题变项的<strong>简单合取式</strong>中，若每个命题变元均以文字的形式出现且仅出现一次，成这样的简单合取式为<strong>极小项</strong>。</p>
<p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323171729779.png" alt="image-20220323171729779"></p>
<p>要使其为真，只需要其中任何一项为真即可。</p>
<p>要使其为假，需要所有项都为假</p>
<p>在含有n个命题变项的<strong>简单析取式</strong>中，若每个命题变元均以文字的形式出现且仅出现一次，成这样的<strong>简单析取式</strong>为<strong>极大项</strong>。</p>
<p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323171721234.png" alt="image-20220323171721234"></p>
<p>若使其为真，则需每个式子都为真</p>
<p>若使其为假，则只需其中的一个式子为假即可</p>
<p><strong>解释</strong>：①文字的形式：p,﹁p,q,﹁q这类的。②仅出现一次：p和﹁p只能有一个出现，即命题变元和其否定只能有一个出现。</p>
<p><strong>说明：</strong></p>
<ul>
<li>n个命题变元可以产生2^n个极小项和2^n个极大项</li>
</ul>
<ol>
<li>极小项（<strong>合取</strong>）：</li>
</ol>
<p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220302221948240.png" alt="image-20220302221948240"></p>
<ul>
<li>没有两个极小项是等价的（即互不等值）</li>
<li>每个极小项只有一组<strong>真</strong>值，因此可用于给极小项编码，<strong>规律</strong>为：命题变元与1对应，其否定与0对应</li>
</ul>
<ol start="2">
<li>极大项（<strong>析取</strong>）：</li>
</ol>
<p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220302222040684.png" alt="image-20220302222040684"></p>
<ul>
<li>没有两个极大项是相同的 </li>
<li>每个极大项只有一组<strong>假</strong>值，因此可用于给极大项编码，规律为：命题变元与0对应，命题变元的否定与1对应</li>
</ul>
<p>3.配凑法求主析取范式和主合取范式</p>
<ul>
<li><p>将析取换成加法，将合取换成乘法，如(p∧q)∨(┐p∧r)代换后变成pq+p’r</p>
</li>
<li><p>求主析取范式&#x3D;&gt;缺少的变元用**乘以x+x’**的形式补充</p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">pq = pq(r + r&#x27;) = pqr+pqr&#x27; (乘以缺失的变元)</span><br><span class="line">p&#x27;r = p&#x27;(q + q&#x27;)r = p&#x27;qr + p&#x27;q&#x27;r</span><br><span class="line">原式 = pqr + pqr&#x27; + p&#x27;qr + p&#x27;q&#x27;r  </span><br></pre></td></tr></table></figure>
</li>
<li><p>换成离散语言<img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323164327707.png" alt="image-20220323164327707"></p>
</li>
<li><p>求主合取范式&#x3D;&gt;将缺少的变元用**加上xx’**的形式补充</p>
<p>里面用到的公式：A+BC&#x3D;(A+B)(A+C)</p>
</li>
<li><pre><code>    pq + p&#39;r                       
 = (p+p&#39;r)(q+p&#39;r)
 = (p+p&#39;)(p+r)(q+r)(q+p&#39;)
 =(p+r)(q+r)(q+p&#39;)
 = (p&#39;+q +rr&#39;)(p+qq&#39;+r)(pp&#39;+q+r)     (补缺的变元)
 = (p&#39;+q+r)(p&#39;+q+r&#39;)(p+q+r)(p+q&#39;+r) (p+q+r)(p&#39;+q +r)   
</code></pre>
<p>换成离散语言：</p>
<p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323164434009.png" alt="image-20220323164434009"></p>
</li>
</ul>
<p>4.将析取范式转化为合取范式</p>
<p>例题一：</p>
<p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220323164512564.png" alt="image-20220323164512564"></p>
<p><strong>实质</strong>：如果将析取看成加法，合取看成乘法，非p用p’表示，则可化为：</p>
<p>(pq’)+(qr)   利用A+BC&#x3D;(A+B)(A+C)</p>
<p>&#x3D;(pq’+q)(pq’+r)</p>
<p>&#x3D;(q+p)(q+q’)(p+r)(q’+r)</p>
<p>例题二：</p>
<p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326172328979.png" alt="image-20220326172328979"></p>
<p><strong>实质</strong>：如果将析取看成加法，合取看成乘法，非p用p’表示，则可化为：</p>
<p>cd+c’d’</p>
<p>&#x3D;(cd+c’)(cd+d’)</p>
<p>&#x3D;(c+c’)(d+c’)(c+d’)(d+d’)</p>
<p>&#x3D;(d+c’)(c+d’)</p>
<h3 id="主范式的用途"><a href="#主范式的用途" class="headerlink" title="主范式的用途"></a>主范式的用途</h3><h4 id="1-求公式的成真赋值和成假赋值"><a href="#1-求公式的成真赋值和成假赋值" class="headerlink" title="1.求公式的成真赋值和成假赋值"></a>1.求公式的成真赋值和成假赋值</h4><p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326173308683.png" alt="image-20220326173308683"></p>
<h4 id="2-判断公式的类型"><a href="#2-判断公式的类型" class="headerlink" title="2.判断公式的类型"></a>2.判断公式的类型</h4><p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326173448446.png" alt="image-20220326173448446"></p>
<p>A为非重言式：A的主析取范式中不含有全部的极小项</p>
<p>A为可满足式：A的主合取范式中不含有全部的极大项</p>
<h4 id="3-判断两个公式是否等值"><a href="#3-判断两个公式是否等值" class="headerlink" title="3.判断两个公式是否等值"></a>3.判断两个公式是否等值</h4><p>若两者的主析取范式或者主合取范式相同，则证明两者等值。</p>
<h4 id="4-解决实际问题"><a href="#4-解决实际问题" class="headerlink" title="4.解决实际问题"></a>4.解决实际问题</h4><p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326173842476.png" alt="image-20220326173842476"></p>
<p>❗解法：</p>
<p>1.将简单命题符号化</p>
<p>2.写出每句话的复合命题</p>
<p>3.写出由②中的复合命题组成的合取式</p>
<p>4.求出③中所得式子的主析取范式</p>
<p>5.找出主析取范式的成真赋值，即为解决方案</p>
<p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326174225108.png" alt="image-20220326174225108"><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326180354192.png" alt="image-20220326180354192">       极小项的成真赋值就是运算的结果</p>
<h3 id="联结词的全功能集"><a href="#联结词的全功能集" class="headerlink" title="联结词的全功能集"></a>联结词的全功能集</h3><h4 id="定义-3"><a href="#定义-3" class="headerlink" title="定义"></a>定义</h4><p>任何命题公式都可以由仅含S中的联结词来表示，则S是联结词全功能集</p>
<p>说明：</p>
<p>若S1是全功能集，则S1中加入其他联结词后构成的S2也是全功能集</p>
<p>若S1不是全功能集，则S1中去掉一些联结词之后构成的S2也不是全功能集</p>
<p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326180948677.png" alt="image-20220326180948677"></p>
<h4 id="复合联结词："><a href="#复合联结词：" class="headerlink" title="复合联结词："></a>复合联结词：</h4><p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326181140453.png" alt="image-20220326181140453"></p>
<p>由此可见，非，或，且联结词都可以用{↑}或者{↓}来表示，说明{↑}，{↓}都是联结词全功能集</p>
<p>注意：</p>
<p>{∨ ， ∧}不是全功能集，因此{∨ }，{∧ }也不是全功能集</p>
<h3 id="组合电路"><a href="#组合电路" class="headerlink" title="组合电路"></a>组合电路</h3><h4 id="逻辑门：与门，非门，或门"><a href="#逻辑门：与门，非门，或门" class="headerlink" title="逻辑门：与门，非门，或门"></a>逻辑门：与门，非门，或门</h4><p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326181608529.png" alt="image-20220326181608529">例题：楼梯的灯由两个开关控制，x,y为开关的状态，F为灯的状态，打开为1，关闭为0，请设计这样一个电路。</p>
<table>
<thead>
<tr>
<th>x</th>
<th>y</th>
<th>F(x,y)</th>
</tr>
</thead>
<tbody><tr>
<td>0</td>
<td>0</td>
<td>1</td>
</tr>
<tr>
<td>0</td>
<td>1</td>
<td>0</td>
</tr>
<tr>
<td>1</td>
<td>0</td>
<td>0</td>
</tr>
<tr>
<td>1</td>
<td>1</td>
<td>1</td>
</tr>
</tbody></table>
<p>由此可见，F的极小项是m0和m3（成真赋值），故根据真值表可以写出其主析取范式，进行化简之后，画出电路图</p>
<p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326182456376.png" alt="image-20220326182456376"></p>
<h3 id="推理理论"><a href="#推理理论" class="headerlink" title="推理理论"></a>推理理论</h3><h4 id="定义-4"><a href="#定义-4" class="headerlink" title="定义"></a>定义</h4><p>对于每组赋值，若满足下列两种情况：①条件为假 ②条件为真，结果为真。则称条件→结论的推理正确，否则推理不正确（即条件为真，结论为假）</p>
<h4 id="判断推理是否正确的方法"><a href="#判断推理是否正确的方法" class="headerlink" title="判断推理是否正确的方法"></a>判断推理是否正确的方法</h4><ul>
<li><p>真值表法</p>
<p>将所有变元可能的取值情况列成真值表，若对于每组赋值，都满足A→B为真，则证明推理正确</p>
</li>
<li><p>等值演算法</p>
<p>将A→B经过等值验算，如果其结果为1，则证明推理正确</p>
</li>
<li><p>主析取范式法</p>
<p>将A→B转化为主析取范式，若其包括了所有的极小项（2^n个），则说名推理正确</p>
</li>
<li><p>构造证明法</p>
</li>
</ul>
<h4 id="推理定律"><a href="#推理定律" class="headerlink" title="推理定律"></a>推理定律</h4><p><img src="/../img/%E7%A6%BB%E6%95%A3%E6%95%B0%E5%AD%A6%E3%81%AE%E7%AC%94%E8%AE%B0/image-20220326183553084.png" alt="image-20220326183553084"></p>

    </div>

    
    
    

      <footer class="post-footer">
          <div class="post-tags">
              <a href="/tags/%E7%AC%94%E8%AE%B0/" rel="tag"># 笔记</a>
          </div>

        


        
    <div class="post-nav">
      <div class="post-nav-item">
    <a href="/2022/03/27/Ajex/" rel="prev" title="Ajex自学笔记">
      <i class="fa fa-chevron-left"></i> Ajex自学笔记
    </a></div>
      <div class="post-nav-item">
    <a href="/2022/04/01/%E6%96%87%E4%BB%B6%E6%93%8D%E4%BD%9C/" rel="next" title="文件操作">
      文件操作 <i class="fa fa-chevron-right"></i>
    </a></div>
    </div>
      </footer>
    
  </article>
  
  
  



          </div>
          

<script>
  window.addEventListener('tabs:register', () => {
    let { activeClass } = CONFIG.comments;
    if (CONFIG.comments.storage) {
      activeClass = localStorage.getItem('comments_active') || activeClass;
    }
    if (activeClass) {
      let activeTab = document.querySelector(`a[href="#comment-${activeClass}"]`);
      if (activeTab) {
        activeTab.click();
      }
    }
  });
  if (CONFIG.comments.storage) {
    window.addEventListener('tabs:click', event => {
      if (!event.target.matches('.tabs-comment .tab-content .tab-pane')) return;
      let commentClass = event.target.classList[1];
      localStorage.setItem('comments_active', commentClass);
    });
  }
</script>

        </div>
          
  
  <div class="toggle sidebar-toggle">
    <span class="toggle-line toggle-line-first"></span>
    <span class="toggle-line toggle-line-middle"></span>
    <span class="toggle-line toggle-line-last"></span>
  </div>

  <aside class="sidebar">
    <div class="sidebar-inner">

      <ul class="sidebar-nav motion-element">
        <li class="sidebar-nav-toc">
          文章目录
        </li>
        <li class="sidebar-nav-overview">
          站点概览
        </li>
      </ul>

      <!--noindex-->
      <div class="post-toc-wrap sidebar-panel">
          <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-3"><a class="nav-link" href="#%E4%B9%A0%E9%A2%98"><span class="nav-number">1.</span> <span class="nav-text">习题</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E5%91%BD%E9%A2%98%E9%80%BB%E8%BE%91"><span class="nav-number">2.</span> <span class="nav-text">命题逻辑</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#%E5%91%BD%E9%A2%98%E4%B8%8E%E8%81%94%E7%BB%93%E8%AF%8D"><span class="nav-number">2.1.</span> <span class="nav-text">命题与联结词</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#%E5%88%A4%E6%96%AD%E7%9C%9F%E5%80%BC"><span class="nav-number">2.2.</span> <span class="nav-text">判断真值</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E4%B8%80%E4%BA%9B%E8%8B%B1%E8%AF%91%E4%B8%AD"><span class="nav-number">3.</span> <span class="nav-text">一些英译中</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E5%91%BD%E9%A2%98"><span class="nav-number">4.</span> <span class="nav-text">命题</span></a><ol class="nav-child"><li class="nav-item nav-level-5"><a class="nav-link" href="#%E6%A6%82%E5%BF%B5%EF%BC%9A"><span class="nav-number">4.0.1.</span> <span class="nav-text">概念：</span></a></li><li class="nav-item nav-level-5"><a class="nav-link" href="#%E4%BE%8B%E9%A2%98%EF%BC%9A"><span class="nav-number">4.0.2.</span> <span class="nav-text">例题：</span></a></li><li class="nav-item nav-level-5"><a class="nav-link" href="#%E5%88%86%E7%B1%BB%EF%BC%9A"><span class="nav-number">4.0.3.</span> <span class="nav-text">分类：</span></a></li></ol></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E5%91%BD%E9%A2%98%E8%81%94%E7%BB%93%E8%AF%8D"><span class="nav-number">5.</span> <span class="nav-text">命题联结词</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E5%91%BD%E9%A2%98%E5%85%AC%E5%BC%8F"><span class="nav-number">6.</span> <span class="nav-text">命题公式</span></a><ol class="nav-child"><li class="nav-item nav-level-5"><a class="nav-link" href="#%E5%AE%9A%E4%B9%89"><span class="nav-number">6.0.1.</span> <span class="nav-text">定义</span></a></li><li class="nav-item nav-level-5"><a class="nav-link" href="#%E7%BA%A6%E5%AE%9A%EF%BC%9A"><span class="nav-number">6.0.2.</span> <span class="nav-text">约定：</span></a></li></ol></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E5%91%BD%E9%A2%98%E7%9A%84%E7%BF%BB%E8%AF%91%E5%92%8C%E7%AC%A6%E5%8F%B7%E5%8C%96"><span class="nav-number">7.</span> <span class="nav-text">命题的翻译和符号化</span></a><ol class="nav-child"><li class="nav-item nav-level-5"><a class="nav-link" href="#%E5%B0%86%E8%87%AA%E7%84%B6%E8%AF%AD%E8%A8%80%E7%AC%A6%E5%8F%B7%E5%8C%96"><span class="nav-number">7.0.1.</span> <span class="nav-text">将自然语言符号化</span></a></li></ol></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E7%9C%9F%E5%80%BC%E8%A1%A8"><span class="nav-number">8.</span> <span class="nav-text">真值表</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E7%AD%89%E4%BB%B7"><span class="nav-number">9.</span> <span class="nav-text">等价</span></a><ol class="nav-child"><li class="nav-item nav-level-5"><a class="nav-link" href="#%E5%AE%9A%E4%B9%89-1"><span class="nav-number">9.0.1.</span> <span class="nav-text">定义</span></a></li><li class="nav-item nav-level-5"><a class="nav-link" href="#%E5%9F%BA%E6%9C%AC%E7%AD%89%E5%80%BC%E5%BC%8F"><span class="nav-number">9.0.2.</span> <span class="nav-text">基本等值式</span></a></li><li class="nav-item nav-level-5"><a class="nav-link" href="#%E7%AD%89%E4%BB%B7%E7%BD%AE%E6%8D%A2%E5%AE%9A%E7%90%86"><span class="nav-number">9.0.3.</span> <span class="nav-text">等价置换定理</span></a></li></ol></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E9%87%8D%E8%A8%80%E5%BC%8F-%E6%B0%B8%E7%9C%9F%E5%BC%8F-%E4%B8%8E%E8%95%B4%E5%90%AB%E5%BC%8F-%E6%B0%B8%E5%81%87%E5%BC%8F"><span class="nav-number">10.</span> <span class="nav-text">重言式(永真式)与蕴含式(永假式)</span></a><ol class="nav-child"><li class="nav-item nav-level-5"><a class="nav-link" href="#%E5%AE%9A%E4%B9%89-2"><span class="nav-number">10.0.1.</span> <span class="nav-text">定义</span></a></li></ol></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E8%8C%83%E5%BC%8F"><span class="nav-number">11.</span> <span class="nav-text">范式</span></a><ol class="nav-child"><li class="nav-item nav-level-5"><a class="nav-link" href="#%E5%AE%9A%E4%B9%89%EF%BC%9A"><span class="nav-number">11.0.1.</span> <span class="nav-text">定义：</span></a></li><li class="nav-item nav-level-5"><a class="nav-link" href="#%E6%B3%A8%E6%84%8F2%EF%BC%9A"><span class="nav-number">11.0.2.</span> <span class="nav-text">注意2：</span></a></li><li class="nav-item nav-level-5"><a class="nav-link" href="#%E8%8C%83%E5%BC%8F%E5%AD%98%E5%9C%A8%E5%AE%9A%E7%90%86"><span class="nav-number">11.0.3.</span> <span class="nav-text">范式存在定理</span></a></li><li class="nav-item nav-level-5"><a class="nav-link" href="#%E8%BD%AC%E5%8C%96%E6%96%B9%E6%B3%95%EF%BC%9A"><span class="nav-number">11.0.4.</span> <span class="nav-text">转化方法：</span></a></li><li class="nav-item nav-level-5"><a class="nav-link" href="#%E4%BE%8B%E9%A2%98%EF%BC%9A-1"><span class="nav-number">11.0.5.</span> <span class="nav-text">例题：</span></a></li></ol></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E4%B8%BB%E8%8C%83%E5%BC%8F"><span class="nav-number">12.</span> <span class="nav-text">主范式</span></a><ol class="nav-child"><li class="nav-item nav-level-5"><a class="nav-link" href="#%E5%8C%96%E6%88%90%E4%B8%BB%E8%8C%83%E5%BC%8F%E7%9A%84%E6%AD%A5%E9%AA%A4%EF%BC%9A"><span class="nav-number">12.0.1.</span> <span class="nav-text">化成主范式的步骤：</span></a></li><li class="nav-item nav-level-5"><a class="nav-link" href="#%E6%9E%81%E5%B0%8F%E9%A1%B9%E5%92%8C%E6%9E%81%E5%A4%A7%E9%A1%B9"><span class="nav-number">12.0.2.</span> <span class="nav-text">极小项和极大项</span></a></li></ol></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E4%B8%BB%E8%8C%83%E5%BC%8F%E7%9A%84%E7%94%A8%E9%80%94"><span class="nav-number">13.</span> <span class="nav-text">主范式的用途</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#1-%E6%B1%82%E5%85%AC%E5%BC%8F%E7%9A%84%E6%88%90%E7%9C%9F%E8%B5%8B%E5%80%BC%E5%92%8C%E6%88%90%E5%81%87%E8%B5%8B%E5%80%BC"><span class="nav-number">13.1.</span> <span class="nav-text">1.求公式的成真赋值和成假赋值</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#2-%E5%88%A4%E6%96%AD%E5%85%AC%E5%BC%8F%E7%9A%84%E7%B1%BB%E5%9E%8B"><span class="nav-number">13.2.</span> <span class="nav-text">2.判断公式的类型</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#3-%E5%88%A4%E6%96%AD%E4%B8%A4%E4%B8%AA%E5%85%AC%E5%BC%8F%E6%98%AF%E5%90%A6%E7%AD%89%E5%80%BC"><span class="nav-number">13.3.</span> <span class="nav-text">3.判断两个公式是否等值</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#4-%E8%A7%A3%E5%86%B3%E5%AE%9E%E9%99%85%E9%97%AE%E9%A2%98"><span class="nav-number">13.4.</span> <span class="nav-text">4.解决实际问题</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E8%81%94%E7%BB%93%E8%AF%8D%E7%9A%84%E5%85%A8%E5%8A%9F%E8%83%BD%E9%9B%86"><span class="nav-number">14.</span> <span class="nav-text">联结词的全功能集</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#%E5%AE%9A%E4%B9%89-3"><span class="nav-number">14.1.</span> <span class="nav-text">定义</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#%E5%A4%8D%E5%90%88%E8%81%94%E7%BB%93%E8%AF%8D%EF%BC%9A"><span class="nav-number">14.2.</span> <span class="nav-text">复合联结词：</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E7%BB%84%E5%90%88%E7%94%B5%E8%B7%AF"><span class="nav-number">15.</span> <span class="nav-text">组合电路</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#%E9%80%BB%E8%BE%91%E9%97%A8%EF%BC%9A%E4%B8%8E%E9%97%A8%EF%BC%8C%E9%9D%9E%E9%97%A8%EF%BC%8C%E6%88%96%E9%97%A8"><span class="nav-number">15.1.</span> <span class="nav-text">逻辑门：与门，非门，或门</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#%E6%8E%A8%E7%90%86%E7%90%86%E8%AE%BA"><span class="nav-number">16.</span> <span class="nav-text">推理理论</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#%E5%AE%9A%E4%B9%89-4"><span class="nav-number">16.1.</span> <span class="nav-text">定义</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#%E5%88%A4%E6%96%AD%E6%8E%A8%E7%90%86%E6%98%AF%E5%90%A6%E6%AD%A3%E7%A1%AE%E7%9A%84%E6%96%B9%E6%B3%95"><span class="nav-number">16.2.</span> <span class="nav-text">判断推理是否正确的方法</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#%E6%8E%A8%E7%90%86%E5%AE%9A%E5%BE%8B"><span class="nav-number">16.3.</span> <span class="nav-text">推理定律</span></a></li></ol></li></ol></div>
      </div>
      <!--/noindex-->

      <div class="site-overview-wrap sidebar-panel">
        <div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person">
    <img class="site-author-image" itemprop="image" alt="芷若"
      src="/images/touxiang.jpg">
  <p class="site-author-name" itemprop="name">芷若</p>
  <div class="site-description" itemprop="description"></div>
</div>
<div class="site-state-wrap motion-element">
  <nav class="site-state">
      <div class="site-state-item site-state-posts">
          <a href="/archives">
          <span class="site-state-item-count">19</span>
          <span class="site-state-item-name">日志</span>
        </a>
      </div>
      <div class="site-state-item site-state-categories">
            <a href="/categories/">
        <span class="site-state-item-count">5</span>
        <span class="site-state-item-name">分类</span></a>
      </div>
      <div class="site-state-item site-state-tags">
            <a href="/tags/">
        <span class="site-state-item-count">7</span>
        <span class="site-state-item-name">标签</span></a>
      </div>
  </nav>
</div>
  <div class="links-of-author motion-element">
      <span class="links-of-author-item">
        <a href="https://github.com/zhiruo" title="GitHub → https:&#x2F;&#x2F;github.com&#x2F;zhiruo" rel="noopener" target="_blank"><i class="fab fa-github fa-fw"></i>GitHub</a>
      </span>
  </div>



      </div>
        <div class="back-to-top motion-element">
          <i class="fa fa-arrow-up"></i>
          <span>0%</span>
        </div>

    </div>
  </aside>
  <div id="sidebar-dimmer"></div>


      </div>
    </main>

    <footer class="footer">
      <div class="footer-inner">
        

        
  <div class="beian"><a href="https://beian.miit.gov.cn/" rel="noopener" target="_blank">陕ICP备2022001874号 </a>
      <img src="http://www.beian.gov.cn/portal/download" style="display: inline-block;">
  </div>

<div class="copyright">
  
  &copy; 
  <span itemprop="copyrightYear">2022</span>
  <span class="with-love">
    <i class="芷若"></i>
  </span>
  <span class="author" itemprop="copyrightHolder">芷若</span>
    <span class="post-meta-divider">|</span>
    <span class="post-meta-item-icon">
      <i class="fa fa-chart-area"></i>
    </span>
      <span class="post-meta-item-text">站点总字数：</span>
    <span title="站点总字数">105k</span>
    <span class="post-meta-divider">|</span>
    <span class="post-meta-item-icon">
      <i class="fa fa-coffee"></i>
    </span>
      <span class="post-meta-item-text">站点阅读时长 &asymp;</span>
    <span title="站点阅读时长">1:35</span>
</div>

        








      </div>
    </footer>
  </div>

  
  <script src="/lib/anime.min.js"></script>
  <script src="/lib/velocity/velocity.min.js"></script>
  <script src="/lib/velocity/velocity.ui.min.js"></script>

<script src="/js/utils.js"></script>

<script src="/js/motion.js"></script>


<script src="/js/schemes/pisces.js"></script>


<script src="/js/next-boot.js"></script>




  




  
<script src="//cdn.jsdelivr.net/npm/algoliasearch@4/dist/algoliasearch-lite.umd.js"></script>
<script src="//cdn.jsdelivr.net/npm/instantsearch.js@4/dist/instantsearch.production.min.js"></script>
<script src="/js/algolia-search.js"></script>














  

  

  

<script src="/live2dw/lib/L2Dwidget.min.js?094cbace49a39548bed64abff5988b05"></script><script>L2Dwidget.init({"pluginRootPath":"live2dw/","pluginJsPath":"lib/","pluginModelPath":"assets/","tagMode":false,"debug":false,"model":{"jsonPath":"/live2dw/assets/assets/koharu.model.json"},"display":{"position":"right","width":145,"height":315},"mobile":{"show":true,"scale":0.5},"react":{"opacityDefault":0.7,"opacityOnHover":0.8},"log":false});</script></body>
</html>
